Using singular values to build a subgrid-scale model for large eddy simulations

Abstract

An eddy-viscosity based, subgrid-scale model for Large Eddy Simulations is derived from the analysis of the singular values of the resolved velocity gradient tensor. The proposed σ-model has by construction the property to automatically vanish as soon as the resolved field is either two-dimensional or two-component, including the pure shear and solid rotation cases. In addition, the model generates no subgrid-scale viscosity when the resolved scales are in pure axisymmetric or isotropic contraction/expansion. At last, it is shown analytically that it has the appropriate cubic behavior in the vicinity of solid boundaries without requiring any ad-hoc treatment. Results for two classical test cases (decaying isotropic turbulence and periodic channel flow) obtained from three different solvers with a variety of numerics (finite elements, finite differences or spectral methods) are presented to illustrate the potential of this model. The results obtained with the proposed model are systematically equivalent or slightly better than the results from the Dynamic Smagorinsky model. Still, the σ-model has a low computational cost, is easy to implement and does not require any homogeneous direction in space or time. It is thus anticipated that it has a high potential for the computation of non-homogeneous, wall-bounded flows.

Image-based large-eddy simulation in a realistic left heart

“…In this work, a SGS model able to handle wall bounded flows in complex geometries is used [54] as well as a well established Dynamic Smagorinsky formulation [55]. Filtering is here performed implicitly, the filter width being equal to the grid spacing.…”

Ventilation of idealised urban area, LES and wind tunnel experiment

“…The nonlinear terms are filtered with filter width equal to two cell widths [15]. The sigma subgrid model [16] is utilised in subfilter (subgrid) scales computations.…”

Building proper invariants for eddy-viscosity subgrid-scale models

“…Notice that τ(u) is considered traceless without the loss of generality, because the trace can be included as part of the filtered pressure, p. Following the same notation as in Nicoud et al, 1 the eddy-viscosity can be modeled in a natural way as follows:…”

“…Most of the existing eddy-viscosity models can be represented by this formulation. 1 Hereafter, in this work, we consider that C m in Eq. (4) is really a constant.…”

“…Isotropic expansions (or contractions) are not considered because they are not compatible with the divergence-free condition, ∇ · u = 0. To complete the list of properties proposed by Nicoud et al, 1 it is also desirable that models are always well-conditioned and have a low (or moderate) computational cost [Property P4]. In this regard, most of them rely on the computation of the invariants of a second-order tensor that depends on the gradient of the resolved velocity field.…”